Relations between the leading terms of a polynomial automorphism

Bonnet, Philippe; Vénéreau, Stéphane

Relations between the leading terms of a polynomial automorphism

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Description

Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal, i.e. $I=(R)$, we compute an upper bound for $°_2(R)$ for some degree function $°_2$ defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of $K^3$ and deduce two elementary proofs of the Jung-van der Kulk Theorem about the tameness of automorphisms of $K^{2}$.
20 pages